An exterior calculus framework for polytopal methods

Francesco Bonaldi (1, 2) , Daniele A. Di Pietro (1) , Jérôme Droniou (3) , Kaibo Hu (4)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations
3 Monash University [Clayton]
4 MI - Mathematical Institute [Oxford]

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

Mots clés: Discrete de Rham complex; Virtual Element Method; differential forms; exterior calculus; Polytopal methods

Domaines: Analyse numérique [math.NA]; Géométrie algébrique [math.AG]; Géométrie différentielle [math.DG]

Francesco Bonaldi, Daniele A. Di Pietro, Jérôme Droniou, Kaibo Hu. An exterior calculus framework for polytopal methods. 2023. ⟨hal-04037653⟩